The backward problem of parabolic equations with the measurements on a discrete set
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Abstract
The backward problems of parabolic equations are of interest in the study of both mathematics and engineering. In this paper, we consider a backward problem for the one-dimensional heat conduction equation with the measurements on a discrete set. The uniqueness for recovering the initial value is proved by the analytic continuation method. We discretize this inverse problem by a finite element method to deduce a severely ill-conditioned linear system of algebra equations. In order to overcome the ill-posedness, we apply the discrete Tikhonov regularization with the generalized cross validation rule to obtain a stable numerical approximation to the initial value. Numerical results for three examples are provided to show the effect of the measurement data.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11771192
Award Identifier / Grant number: 11971121
Funding statement: The work described in this paper was supported by a China NSF grant (No.11771192, 11971121).
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Articles in the same Issue
- Frontmatter
- Modified Radon transform inversion using moments
- Inverse source problem for a distributed-order time fractional diffusion equation
- Two closed novel formulas for the generalized inverse A T,S (2) of a complex matrix with given rank
- The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type
- Inverse problems for a class of linear Sobolev type equations with overdetermination on the kernel of operator at the derivative
- Isospectral sets for transmission eigenvalue problem
- Stability estimate for an inverse problem of the convection-diffusion equation
- The enclosure method for the heat equation using time-reversal invariance for a wave equation
- Joint inversion of compact operators
- Inverse problem of breaking line identification by shape optimization
- The backward problem of parabolic equations with the measurements on a discrete set
- Optimal convergence rates for inexact Newton regularization with CG as inner iteration
Articles in the same Issue
- Frontmatter
- Modified Radon transform inversion using moments
- Inverse source problem for a distributed-order time fractional diffusion equation
- Two closed novel formulas for the generalized inverse A T,S (2) of a complex matrix with given rank
- The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type
- Inverse problems for a class of linear Sobolev type equations with overdetermination on the kernel of operator at the derivative
- Isospectral sets for transmission eigenvalue problem
- Stability estimate for an inverse problem of the convection-diffusion equation
- The enclosure method for the heat equation using time-reversal invariance for a wave equation
- Joint inversion of compact operators
- Inverse problem of breaking line identification by shape optimization
- The backward problem of parabolic equations with the measurements on a discrete set
- Optimal convergence rates for inexact Newton regularization with CG as inner iteration
